please help !! Given l, m, n, find the value of x.
(7x-2)° (5x+14)°

Answer: approximately 68,404 liters.

Step-by-step explanation:

To solve the problem, we first need to find the cross-sectional area of the pipe, which we can calculate using the formula for the area of a circle:

A = πr^2

where r is the radius of the pipe, which is half the diameter. So, in this case, the radius is 2.4 cm / 2 = 1.2 cm.

A = π(1.2 cm)^2

A ≈ 4.5239 cm^2

Next, we can use the formula for volume flow rate to find the volume of water that is discharged per second:

Q = Av

where Q is the volume flow rate, A is the cross-sectional area of the pipe, and v is the velocity of the water. In this case, we have:

Q = (4.5239 cm^2)(2.8 m/s)

Q ≈ 0.01266 m^3/s

To find the volume of water discharged in one and a half hours (which is 5400 seconds), we can multiply the volume flow rate by the time:

V = Qt

V = (0.01266 m^3/s)(5400 s)

V ≈ 68.404 m^3

Finally, to convert the volume from cubic meters to liters, we can multiply by 1000:

V = 68.404 m^3 × 1000 L/m^3

V ≈ 68,404 L

Therefore, the volume of water discharged in one and a half hours is approximately 68,404 liters.

For the given polynomials in factored form;

(a) the leading term is x³, the zeroes are -2,3 and 5 , the graph will be a cubic function passing through x-axis at -2, 3 and 5.
(b) the leading term is 2x², the zeros are -1, 4 the graph will be a quadratic function passing through x-axis at -1 and 4.

Part(a) : The polynomial in factored form is : f(x) = (x + 2)(x - 3)(x - 5)

The Leading Term is : x³; The Zeros are : -2, 3, 5.

The General Shape: The graph of the polynomial will be a cubic function that passes through the x-axis at -2, 3, and 5.

The function will approach negative infinity as x approaches negative infinity and positive infinity as x approaches positive infinity.

Part(b) : The Polynomial in factored form is : f(x) = 2(x + 1)(x - 4)

The Leading Term is : 2x²;  The Zeros are : -1, 4.

The General Shape: The graph of the polynomial will be a quadratic function that passes through the x-axis at -1 and 4. The function will open upwards since the leading coefficient is positive.

The function will approach negative infinity as x approaches negative infinity and positive infinity as x approaches positive infinity.

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The given question is incomplete, the complete question is

For each polynomial in factored form show the leading term, the zeros on the x-axis, and the general shape of the polynomial.

(a) f(x) = (x + 2)(x - 3)(x - 5)

(b) f(x) = 2(x + 1)(x - 4).

The scale of the map in the question is 1 inch = 50 miles.

A scale in a map is a relation that tells us how many units each unit in the map represents. In this case, we know that the distance between two points A and B on the map is 5 inches, while the actual distance between these two places is 250 miles.

Then we start with the relation:

5 inches = 250 miles.

But to get the scale of the map we need to see how many miles one inch represents in the map, then we can divide both sides of the equation by 5 to geT:

5 in = 250 mi

1 in = 250mi/5

1 in = 50 mi

The scale of the map is 1 inch to 50 miles.

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Answer:

The numbers 8 and lower are possible values of x.

Step-by-step explanation:

The inequality [tex]x\leq 8[/tex] means x is less than or equal to 8. Therefore, 8 is a possible value of x.

If a is a zero of the polynomial P(x), then (x-a) must be a factor of P(x) and [tex](x-a)^m[/tex] be a factor of P(x) when we factor P completely.

The values of x that fulfil the formula f(x) = 0 are the zeros of a polynomial. The polynomial's zeros are the x values for which the function's value, f(x), equals zero in this case. The degree of the equation f(x) = 0 determines how many zeros a polynomial has.

The locations when a polynomial equals 0 overall are known as its zeros. In layman's terms, we may state that a polynomial's zeros are variable values at which the polynomial equals 0. The zeros of a polynomial are often referred to as the equation's roots and are frequently written as,, and. A few techniques for locating polynomial zeros include grouping, factoring, and employing algebraic expressions.

(a) if we have zero at x=a of polynomial P(x)

then,  (x-a)  must be factor of P(x).

(b) if we have zero at x=a of polynomial P(x)

with multiplicity=m

then,  [tex](x-a)^m[/tex]  must be factor of P(x).

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Given that (1, 2, 3] System{1, 4, 7,6] for a system known to be LTI, the impulse response of the system: h[n] = (1/2)*δ[n] + δ[n-1] + (3/2)*δ[n-2]

To compute the impulse response h[n] of a linear time-invariant (LTI) system given its input-output relationship, we can use the convolution sum:

y[n] = x[n] * h[n]

y[n] = (1/2)*(x[n] + 2x[n-1] + 3x[n-2])

y[n] = (1/2)*(δ[n] + 2δ[n-1] + 3δ[n-2])

y[n] = (1/2)*δ[n] + δ[n-1] + (3/2)*δ[n-2]

Thus, the impulse response of the system is:

h[n] = (1/2)*δ[n] + δ[n-1] + (3/2)*δ[n-2],where δ[n] is the impulse signal.

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The equation is saying that -12 is equal to -3 multiplied by y. To solve for y, divide both sides by -3. This would give an answer of 4.

An equation is a mathematical statement that expresses the equality or inequality of two values or expressions. It consists of two expressions connected by an equals sign, inequality sign or other relational operator. Equations can involve numbers, variables, and operations such as addition, subtraction, multiplication, division and exponentiation. An equation can be used to solve problems related to mathematics, science, engineering, finance, and many other disciplines. Equations can also be used to model and describe real-world phenomena.

t = 30

a = 12.5

p = -5.5

x = 2/3

y = 4.

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a. t = 30/2; To solve this equation, divide both sides by 2/5. The resulting equation is t = 30/2.

An equation is a mathematical statement that expresses the equality of two expressions by using an equals sign (=). It states that the two expressions on either side of the equals sign are equal in value. An equation is an example of a mathematical problem, which can be used to solve real-world problems.

b. a = 4.5; To solve this equation, add 8 to both sides. The resulting equation is a = 4.5.
c. p = -7/2; To solve this equation, add 3 to both sides. The resulting equation is p = -7/2.
d. x = 2; To solve this equation, divide both sides by 3. The resulting equation is x = 2.
e. y = 4; To solve this equation, divide both sides by -3. The resulting equation is y = 4.

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Step-by-step explanation:

Area of parallelogram = base × height or say b×h

Given: h=8 inches

Area=120 sq.inches

⇒120=8×b

⇒120/8=b

⇒120/8=b = 15

Answer = 15 inches b

Answer:

$31122.00

Step-by-step explanation:

We know

She works 35 hours a week, earning $17.10 an hour.

17.10 x 35 = $598.50 a week

How much does she earn in one year?

We Take

598.50 x 52 = $31122.00

So, she earns $31122.00 one year.

The expression's fully factored form is:[tex]72x^{3} + 72x^{2} + 18x = 18x(4x^{2} + 1)(x + 1)[/tex]

Factored Value, also known as "trended value," is the base annual value plus a yearly inflation factor based on a variation in the cost if live that is not to exceed 2% and is set by the State Agency of Equalization.

Rewriting an expression as the sum of factors is referred to as factor expressions or factoring. For instance, 3x + 12y may be expressed as 3 (x + 4y), which is a straightforward equation. The computations get simpler in this method. Three or (x + 4y) were examples of factors.

We can factor out [tex]18x[/tex] from each term to simplify the expression:

[tex]72x^{3} + 72x^{2} + 18x = 18x(4x^{3} + 4x^{2} + 1)[/tex]

An expression enclosed in parentheses can now be calculated by grouping or factoring.

[tex]4x^{3} + 4x^{2} + 1 = (4x^{2} + 1)(x + 1)[/tex]

The expression's properly factored version has the following result,

[tex]72x^{3} + 72x^{2} + 18x = 18x(4x^{2} + 1)(x + 1)[/tex]

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Answer: 44 miles

Step-by-step explanation:

Let's call the number of miles Kevin can travel "m". We can set up an equation using the given information:

4.25 + 0.47m ≤ 25

Solving for "m", we can begin by subtracting 4.25 from both sides:

0.47m ≤ 20.75

Then, divide both sides by 0.47:

m ≤ 44.15

Therefore, Kevin can travel at most 44 miles without exceeding his spending limit.

All answers are mentioned below.

A balance sheet is a financial statement that provides a snapshot of a company's financial position at a specific point in time. It presents the company's assets, liabilities, and equity, and shows how these are financed. The balance sheet follows the accounting equation, which states that the total assets must equal the sum of liabilities and equity.

(a) Trading, Profit & Loss and Profit & Loss Appropriation Account for the year ended December 31, 1996

Trading Account

$

Sales (51,550 - 5,550 + 5,385) 51,385

Less: Cost of Goods Sold

Opening Stock 5,550

Add: Purchases 29,525

35,075

Less: Closing Stock 5,385

Cost of Goods Sold 29,690

Gross Profit 21,695

Profit & Loss Account

$

Gross Profit 21,695

Less: Expenses

Carriage Inward 100

Discount Allowed 1,000

Wages & Salaries 17,650

Motor Vehicle Repairs 5,850

Telephones 350

Rent & Rates 1,800

Advertising 2,750

Carriage Outward 1,000

Bank Charges 550

Depreciation

Land & Building (30,000/50 years) 600

Motor Vehicle (20% x 20,000) 4,000

34,550

Net Loss (12,855)

Profit & Loss Appropriation Account

$

Net Loss (12,855)

Add: Interest on Capital (5% x $90,000) 4,500

(8,355)

To be shared equally between Knox and Cox

Knox (4,178)

Cox (4,178)

Additional Salary for Cox (5,000)

Carried Forward (13,356)

(b) Partners' Current Account

Knox Cox

$ $

To Opening Balance 8,185 4,400

By Share of Net Loss (4,178) (4,178)

By Interest on Capital 2,500 2,000

By Drawings (1,200) (500)

By Salary 0 (5,000)

To Closing Balance 5,307 (3,278)

(c) Balance Sheet as at December 31, 1996

Fixed Assets

Land & Building (100,000 - 30,000) 70,000

Motor Vehicles (20,000 - 4,000) 16,000

86,000

Current Assets

Stock 5,385

Debtors 7,250

Bank 3,200

Cash 510

16,345

Current Liabilities

Creditors (7,650)

Net Current Assets 8,695

94,695

Less: Partners' Capital and Current Accounts

Knox Capital (50,000 + 5,307) 55,307

Cox Capital (40,000 - 3,278) 36,722

92,029

2,666

Notes:

Depreciation of Land and Building: 30,000/50 years = 600 per year

Depreciation of Motor Vehicles: 20% x 20,000 = 4,000

Interest on Capital: Knox - 5% x 50,000 = 2,500, Cox - 5% x 40,000 = 2,000

Interest on Drawings: Knox - 10% x 1,200 = 120, Cox - 10% x 500 = 50

Additional Salary for Cox is treated as an expense and charged to the Profit & Loss Account.

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(a) Trading, Profit & Loss and Profit for the year ended December 31, 1996 is mentioned below. (b) Knox Cox, Opening Balance 8,185 4,400, respectively.(c) Interest on Drawings: Knox - 10% x 1,200 = 120, Cox - 10% x 500 = 50

A balance sheet is a financial statement that provides a snapshot of a company's financial position at a specific point in time. It presents the company's assets, liabilities, and equity, and shows how these are financed. The balance sheet follows the accounting equation, which states that the total assets must equal the sum of liabilities and equity.

(a) Trading, Profit & Loss and Profit & Loss Appropriation Account for the year ended December 31, 1996

Trading Account

$

Sales (51,550 - 5,550 + 5,385) 51,385

Less: Cost of Goods Sold

Opening Stock 5,550

Add: Purchases 29,525

35,075

Less: Closing Stock 5,385

Cost of Goods Sold 29,690

Gross Profit 21,695

Profit & Loss Account

$

Gross Profit 21,695

Less: Expenses

Carriage Inward 100

Discount Allowed 1,000

Wages & Salaries 17,650

Motor Vehicle Repairs 5,850

Telephones 350

Rent & Rates 1,800

Advertising 2,750

Carriage Outward 1,000

Bank Charges 550

Depreciation

Land & Building (30,000/50 years) 600

Motor Vehicle (20% x 20,000) 4,000

34,550

Net Loss (12,855)

Profit & Loss Appropriation Account

Net Loss (12,855)

Add: Interest on Capital (5% x $90,000) 4,500

(8,355)

To be shared equally between Knox and Cox

Knox (4,178)

Cox (4,178)

Additional Salary for Cox (5,000)

Carried Forward (13,356)

(b) Partners' Current Account

                                    Knox Cox

                                          $ $

To Opening Balance 8,185 4,400

By Share of Net Loss (4,178) (4,178)

By Interest on Capital 2,500 2,000

By Drawings               (1,200) (500)

By Salary                         0 (5,000)

To Closing Balance     5,307 (3,278)

(c) Balance Sheet as at December 31, 1996

Fixed Assets

Land & Building (100,000 - 30,000) 70,000

Motor Vehicles (20,000 - 4,000) 16,000

86,000

Current Assets

Stock 5,385

Debtors 7,250

Bank 3,200

Cash 510

16,345

Current Liabilities

Creditors (7,650)

Net Current Assets 8,695

94,695

Less: Partners' Capital and Current Accounts

Knox Capital (50,000 + 5,307) 55,307

Cox Capital (40,000 - 3,278) 36,722

92,029

2,666

Notes:

Depreciation of Land and Building: 30,000/50 years = 600 per year

Depreciation of Motor Vehicles: 20% x 20,000 = 4,000

Interest on Capital: Knox - 5% x 50,000 = 2,500, Cox - 5% x 40,000 = 2,000

Interest on Drawings: Knox - 10% x 1,200 = 120, Cox - 10% x 500 = 50

Additional Salary for Cox is treated as an expense and charged to the Profit & Loss Account.

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Answer:

2.5 hours

Step-by-step explanation:

To find the number of hours for each test,

Divide the number of hours by the number of tests

10 divided by 4

2.5 hours for each test

Answer:

Step-by-step explanation:

True. 1 is an odd positive integer less than 11 and is an element of set B.

The symmetric equations of the line passing through a point and parallel to a vector are -(x - 4) = y + 1/4 = -(z - 9)/2. The line intersects the xy-, xz-, and yz-planes at (5, -9/4, 0), (15/4, 0, 23/2), and (0, -17/4, 11/2), respectively.

To find the symmetric equations of the line, we first need to find the direction vector of the line. Since the line is parallel to the vector <4, -1, 9>, any scalar multiple of this vector will be a direction vector of the line. So, let's choose the parameter t and write the vector equation of the line:

r = <4, -1, 9> + t<-1, 4, -2>

Expanding this vector equation component-wise, we get:

x = 4 - t

y = -1 + 4t

z = 9 - 2t

These equations can be rearranged to get the symmetric equations of the line:

-(x - 4) = y + 1/4 = -(z - 9)/2

To find the points in which the line intersects the coordinate planes, we substitute the corresponding variables with 0 in the equations for the line.

For the xy-plane, we set z = 0 and solve for x and y:

-(x - 4) = y + 1/4 = -(-9)/2

x = 5, y = -9/4

So, the line intersects the xy-plane at the point (5, -9/4, 0).

For the xz-plane, we set y = 0 and solve for x and z:

-(x - 4) = 0 + 1/4 = -(z - 9)/2

x = 15/4, z = 23/2

So, the line intersects the xz-plane at the point (15/4, 0, 23/2).

For the yz-plane, we set x = 0 and solve for y and z:

-(-4) = y + 1/4 = -(z - 9)/2

y = -17/4, z = 11/2

So, the line intersects the yz-plane at the point (0, -17/4, 11/2).

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Answer: Questionnaire refers to a research instrument, in which a series of question, is typed or printed along with the choice of answers, expected to be marked by the respondents, used for survey or statistical study. It consists of aformalisedd set of questions, in a definite order on a form, which are mailed to the respondents or manually delivered to them for answers. The respondents are supposed to read, comprehend and give their responses, in the space provided.

A ‘Pilot Study’ is advised to be conducted to test the questionnaire before using this method. A pilot survey is nothing but a preliminary study or say rehearsal to know the time, cost, efforts, reliability and so forth involved in it.

The interview is a data collection method wherein a direct, in-depth conversation between interviewer and respondent takes place. It is carried out with a purpose like a survey, research, and the like, where both the two parties participate in the one to one interaction. Under this method, oral-verbal stimuli are presented and replied by way of oral-verbal responses.

It is considered as one of the best methods for collecting data because it allows two way exchange of information, the interviewer gets to know about the respondent, and the respondent learns about the interviewer. There are two types of interview:

Personal Interview: A type of interview, wherein there is a face to face question-answer session between the interviewer and interviewee, is conducted.

Telephonic Interview: This method involves contacting the interviewee and asking questions to them on the telephone itself.

Rumiya's total earnings can be represented by the inequality: [tex]85000 + 0.1x > 100000[/tex] and she would need to make sales of at least $150,000 to earn over $100,000 for the year.

What do you mean by commission and inequality ?

A commission is a percentage of sales that a salesperson earns on top of their base salary. In this case, Rumiya earns a 10% commission on sales she makes for the year. An inequality is a statement that compares two values, indicating whether one is greater than, less than, or equal to the other. It is used to represent that Rumiya needs to make sales that exceed a certain amount in order to earn a desired amount.

Finding the minimum amount of sales :

Rumiya's total earnings for the year will be the sum of her base salary and commission on sales. We can represent this as an inequality:

[tex]85000 + 0.1x > 100000[/tex]

To solve for [tex]x[/tex], we first need to isolate the variable on one side of the inequality. We can do this by subtracting 85000 from both sides:

[tex]0.1x > 15000[/tex]

Next, we can solve for [tex]x[/tex] by dividing both sides by 0.1:

[tex]x > 150000[/tex]

Therefore, Rumiya would need to make sales of at least $150,000 to earn over $100,000 for the year. This means that her commission on these sales would be $15,000 (10% of $150,000).

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Answer:

Step-by-step explanation:

Let x be the amount invested in the first account, which pays 14% interest. Then the amount invested in the second account, which pays 11% interest, is 6800 - x.

The interest earned on the first account is 0.14x, and the interest earned on the second account is 0.11(6800 - x). The total interest earned is the sum of these two amounts, so we have:

0.14x + 0.11(6800 - x) = 856

Simplifying and solving for x, we get:

0.14x + 748 - 0.11x = 856

0.03x = 108

x = 3600

Therefore, Asaad invested $3600 in the first account and $3200 (6800 - 3600) in the second account.

Answering the presented question, we may conclude that This greater expressions scale makes it easier to see the variations between the ratings of the three college students in a extra correct and informative way.

In mathematics, an expression is a collection of integers, variables, and complex mathematical (such as arithmetic, subtraction, multiplication, division, multiplications, and so on) that describes a quantity or value. Phrases can be simple, such as "3 + 4," or complicated, such as They may also contain functions like "sin(x)" or "log(y)". Expressions can be evaluated by swapping the variables with their values and performing the arithmetic operations in the order specified. If x = 2, for example, the formula "3x + 5" equals 3(2) + 5 = 11. Expressions are commonly used in mathematics to describe real-world situations, construct equations, and simplify complicated mathematical topics.

The difference in look between the two charts is due to the choice of the scales on the x and y-axes. In the left chart, the y-axis starts offevolved at 800 and has a range of only 200 points, whilst the x-axis starts offevolved at 1300 and has a vary of 200 points. This compressed scale makes the variations between the ratings of the three students appear larger than they absolutely are.

On the other hand, the proper chart has a y-axis that starts at zero and has a vary of 800 points, whilst the x-axis begins at 1200 and has a range of 800 points. This greater expanded scale makes it easier to see the variations between the ratings of the three college students in a extra correct and informative way.

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Probability of winning at home is 0.67, Probability of winning away is 0.58.

To calculate the probability of home and away wins, we need to divide the number of wins by the total number of games played at home and away respectively.

Probability of winning at home = (Number of wins at home) / (Total number of games played at home)

= 8 / 12

= 0.67

Probability of winning away = (Number of wins away) / (Total number of games played away)

= 7 / 12

= 0.58

Based on these results, Sasha's team has a higher probability of winning at home (0.67) compared to away games (0.58).

Therefore, her belief that her team plays better away is not supported by the data.

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The expression (2^8) to the power of 3 as a power of 2 is  4096^2

Given the following expression

(2^8) to the power of 3

To write (2^8) to the power of 3 as a power of 2, we can use the rule of exponents that says:

(a^b)^c = (a^c)^b

Applying this rule to (2^8)^3, we get:

(2^8)^3 = (2^3)^8

Simplifying the expression, we get:

(2^8)^3 = (8)^8

So, we have

(2^8)^3 = (8^4)^2

Simplify

(2^8)^3 = 4096^2

Therefore, (2^8)^3 can be written as 4096^2.

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Answer:

Step-by-step explanation:

The surface area of the resulting solid is 4πr².

The surface area of a circle of radius r centered at (r,0), rotated about the y-axis, can be determined by first finding the area of the circle and then adding the area of the cylinder formed by rotating the circle about the y-axis.

The area of the circle is given by A = πr².

The area of the cylinder formed by rotating the circle about the y-axis is equal to the circumference of the circle (2πr) multiplied by the height of the cylinder (2r). Therefore, the total surface area of the rotated circle is equal to the area of the circle (πr²) plus the area of the cylinder (2πr * 2r) which gives a total surface area of 4πr².

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Answer:

4

Step-by-step explanation:

One pair of functions that satisfies the given condition is:

Exponential function: [tex]f(x) = 1.46^x,[/tex] Quadratic function: [tex]g(x) = x^2[/tex]

In mathematics, an expression is a combination of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. Expressions can also include functions, brackets, and other symbols.

Let's consider the two functions:

Exponential function: [tex]f(x) = a^x, where a > 1[/tex]

Quadratic function: [tex]g(x) = x^2[/tex]

We want to find the pair of functions for which the exponential is consistently growing at a faster rate than the quadratic over the interval 0 ≤ x ≤ 5.

To determine this, we can compare the growth rates of the two functions by looking at their derivatives.

The derivative of the exponential function is:[tex]f'(x) = a^x * ln(a)[/tex]

The derivative of the quadratic function is: [tex]g'(x) = 2x[/tex]

To compare the growth rates of the two functions, we need to compare their derivatives. We want to find the value of x for which the exponential function is growing faster than the quadratic function, i.e., where f'(x) > [tex]g'(x).\\f'(x) > g'(x)\\a^x * ln(a) > 2x[/tex]

Now, we can solve for x:

[tex]a^x * ln(a) > 2xln(a)/2 * a^x > x[/tex]

Since we want to find the pair of functions for which the exponential is consistently growing at a faster rate than the quadratic over the interval 0 ≤ x ≤ 5, we need to find a value of a such that the inequality ln(a)/2 * [tex]a^5 > 5[/tex] is true for all values of a > 1.

We can use a graphing calculator or a numerical solver to find the value of a that satisfies this inequality. One possible solution is a ≈ 1.46.

Therefore, one pair of functions that satisfies the given condition is:

Exponential function: [tex]f(x) = 1.46^x,[/tex] Quadratic function: [tex]g(x) = x^2[/tex]

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The answer of the given question based on solving  13 to the power of 16 is 13 to the power of 16 is 3,947,868,257,259,789. and the simplified expression is 7 + 2ab + br.

A expression is a combination of symbols or values that represents a particular concept or computation.

In mathematics, an expression is a combination of numbers, variables, operators, and/or functions that can be evaluated to produce a numerical result.

an expression is a way to represent an idea, computation, or meaning using a set of symbols or words.

To solve 13 to the power of 16, we can start by multiplying 13 by itself 16 times:

13 × 13 = 169

169 × 13 = 2197

2197 × 13 = 28,561

28,561 × 13 = 371,293

371,293 × 13 = 4,826,389

4,826,389 × 13 = 62,748,857

62,748,857 × 13 = 815,730,721

815,730,721 × 13 = 10,604,807,473

10,604,807,473 × 13 = 137,858,491,849

137,858,491,849 × 13 = 1,792,160,390,737

1,792,160,390,737 × 13 = 23,303,986,079,681

23,303,986,079,681 × 13 = 303,305,489,096,753

303,305,489,096,753 × 13 = 3,947,868,257,259,789

Therefore, 13 to the power of 16 is 3,947,868,257,259,789.

As for the second expression, we can simplify it using the distributive property of multiplication:

2(2 + ab) + b(r + 3) = 4 + 2ab + br + 3b

Simplifying further, we can combine the constant terms:

2(2 + ab) + b(r + 3) = 7 + 2ab + br

So the simplified expression is 7 + 2ab + br.

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The required answers are 1) [tex]$$A = x^2 + 28x + 192$$[/tex] 2) 300000 3) [tex]$$x^2 + 28x + 192 = (x + 14 - 2\sqrt{19})(x + 14 + 2\sqrt{19})$$[/tex].

area of the land purchased is given as 480m², and the dimensions of the land are (x+12)mx(x+16)m. Therefore, the area of the land can be expressed as:

[tex]$$A = (x+12)(x+16)$$[/tex]

Expanding this expression, we get:

[tex]$$A = x^2 + 28x + 192$$[/tex]

Hence, the area of the land purchased is given by the polynomial expression [tex]$x^2 + 28x + 192$[/tex].

The total budget for the expenses is Rs. 5,00,000. If Mr. Chand's daughter is ready to share 3/5 of the expenses, then the fraction of the expenses she will pay is:

[tex]$\frac{3}{5}=\frac{x}{500000}$$[/tex]

Simplifying this expression, we get:

[tex]$x = \frac{3}{5}\times 500000 = 300000$$[/tex]

Therefore, Mr. Chand's daughter will pay Rs. 3,00,000 towards the expenses.

We can solve the polynomial [tex]$x^2 + 28x + 192$[/tex] into different factors by using the quadratic formula:

[tex]$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$[/tex]

Here, the coefficients of the polynomial are:

[tex]$$a = 1, \quad b = 28, \quad c = 192$$[/tex]

Substituting these values in the quadratic formula, we get:

[tex]$x = \frac{-28 \pm \sqrt{28^2 - 4\times 1 \times 192}}{2\times 1}$$[/tex]

Simplifying this expression, we get:

[tex]$$x = -14 \pm 2\sqrt{19}$$[/tex]

Therefore, the polynomial [tex]$x^2 + 28x + 192$[/tex] can be factored as:

[tex]$$x^2 + 28x + 192 = (x - (-14 + 2\sqrt{19}))(x - (-14 - 2\sqrt{19}))$$[/tex]

or

[tex]$$x^2 + 28x + 192 = (x + 14 - 2\sqrt{19})(x + 14 + 2\sqrt{19})$$[/tex]

So, we have factored the polynomial into two factors.

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Answer:

[tex]5x+11[/tex]

Step-by-step explanation:

Step 1: Distribute

[tex]3x+6+2x+5[/tex]

Step 2: Add like terms

[tex]5x+11[/tex] < your answer

The required probabilities of catching or not catching a fish is given by,

Catching fish in all 4 lakes = 0.0048

Not catching any fish at all = 0.2688

Catching at least one fish  = 0.7312

Catching fish in Lake L1 and lake L2 = 0.12

Catching fish in lake L1 or lake L2 = 0.58

Catching fish in exactly one lake = 1.1

Probability of catching fish in all 4 lakes,

Simply multiply the probabilities of catching fish in each lake,

since the events are independent,

P(C1 and C2 and C3 and C4)

= P(C1) x P(C2) x P(C3) x P(C4)

= 0.3 x 0.4 x 0.2 x 0.2

= 0.0048

= 0.48%

Probability of not catching any fish at all,

Probability of the complement of the event of catching at least one fish.

Happy if we catch at least one fish, the probability of not catching any fish is the probability that is not happy,

P(not happy)

= P(not C1 and not C2 and not C3 and not C4)

= (1 - P(C1)) x (1 - P(C2)) x (1 - P(C3)) x (1 - P(C4))

= 0.7 x 0.6 x 0.8 x 0.8

= 0.2688

= 26.88%

Probability of catching at least one fish,

Probability of the complement of the event of not catching any fish and subtract it from 1,

P(at least one fish)

= 1 - P(not happy)

= 1 - 0.2688

= 0.7312

= 73.12%

Probability of catching fish in Lake L1 and lake L2,

Simply multiply the probabilities of catching fish in each lake,

P(C1 and C2)

= P(C1) x P(C2)

= 0.3 x 0.4

= 0.12

= 12%

Probability of catching fish in lake L1 or lake L2,

Add the probabilities of catching fish in each lake,

And then subtract the probability of catching fish in both lakes to avoid double counting,

P(C1 or C2)

= P(C1) + P(C2) - P(C1 and C2)

= 0.3 + 0.4 - 0.12

= 0.58

= 58%

Probability of catching fish in exactly one lake can be broken down into four mutually exclusive events,

Catching fish in L1 only, catching fish in L2 only, catching fish in L3 only, or catching fish in L4 only.

Probabilities of each of these events is,

P(C1 or C2 or C3 or C4)

= P(C1) +  P(C2) + P(C3) + P(C4)

=  0.3 + 0.4 + 0.2 + 0.2

= 1.1

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